Question

a t distribution with 11 degrees of freedom is graphed below. the region under the curve to the right of $t_{0.97}$ is shaded. the area of this region is 0.97. find the value of $t_{0.97}$. round your answer to three decimal places. $t_{0.97}=\square$

Explanation:

Step1: Recall the property of t - distribution

The total area under the t - distribution curve is 1. If the area to the right of \(t_{0.97}\) is 0.97, then the area to the left of \(t_{0.97}\) is \(1 - 0.97=0.03\).

Step2: Use t - distribution table or calculator

We use a t - distribution table or a calculator with a t - distribution function. For a t - distribution with \(n = 11\) degrees of freedom, we look for the value of \(t\) such that \(P(T<t)=0.03\) (where \(T\) follows a t - distribution with \(n = 11\) degrees of freedom).
Using a calculator (for example, in R: qt(0.03, 11)), we get the value.

Answer:

\(- 1.998\)